Inferring Latent States and Refining Force Estimates via Hierarchical

1
arXiv:1501.00303v2 [q-bio.QM] 5 Nov 2015
Inferring Latent States and Refining Force Estimates via
Hierarchical Dirichlet Process Modeling in Single Particle
Tracking Experiments †
Christopher P. Calderon1,∗ , Kerry Bloom 2 ,
1 Ursa Analytics, Denver, CO, USA
2 Department of Biology, University of North Carolina, Chapel Hill, NC, USA
∗ E-mail: Corresponding Chris.Calderon@UrsaAnalytics.com
Abstract
Understanding the basis for intracellular motion is critical as the field moves toward a deeper understanding
of the relation between Brownian forces, molecular crowding, and anisotropic (or isotropic) energetic forcing.
Effective forces and other parameters used to summarize molecular motion change over time in live cells
due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and
other heterogeneity inherent in biological processes. This study discusses limitations in currently popular
analysis methods (e.g., mean square displacement-based analyses) and how new techniques can be used to
systematically analyze Single Particle Tracking (SPT) data experiencing abrupt state changes in time or
space. The approach is to track GFP tagged chromatids in metaphase in live yeast cells and quantitatively
probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical
phenomena. State changes can be induced by various sources including: microtubule dynamics exerting
force through the centromere, thermal polymer fluctuations, and DNA-based molecular machines including
polymerases and protein exchange complexes such as chaperones and chromatin remodeling complexes.
Simulations aiming to show the relevance of the approach to more general SPT data analyses are also
studied. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian
modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS),
for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical
regime changes induced by unobserved state changes when the number of underlying states is unknown
in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS
approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete
time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we
demonstrate new computational techniques for tuning hyperparameters and for checking the statistical
consistency of model assumptions directly against individual experimental trajectories; the techniques
circumvent the need for “ground-truth” and/or subjective information.
† Differs
only typographically from PLoS One article available freely as an open-access article at:
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137633
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Introduction
Recent advances in optical microscopy [1–16] have inspired numerous analysis methods aiming to quantify
the motion of individual molecules in live cells [17–29]. The resolution afforded by current optical
microscopes allows researchers to more reliably measure two-dimensional (2D) [17, 27, 28] and threedimensional (3D) [23] position vs. time data in Single Particle Tracking (SPT) experiments. This permits
researchers to probe in vivo forces without introducing external perturbations into the system. Techniques
capable of reliably quantifying the in vivo forces experienced by single-molecules (without ensemble
averaging) offer the potential to gain new molecular-level understanding of various complex biological
processes including cell division [24], virus assembly [30], endocytosis [31] and drug delivery [15].
In this article, we demonstrate how the Hierarchical Dirichlet Process Switching Linear Dynamical
System (HDP-SLDS) framework developed by Fox and co-workers [32] can be used to deduce the direction
and magnitude of different forces that contribute to molecular motion in living cells [23]. The utility of
combining the HDP-SLDS with SPT was motivated by experiments aiming to quantify the time varying
forces driving chromosome dynamics. The approach presented shows promise in both (I) accelerating the
scientific discovery process (i.e., statistically significant changes in dynamics can be reliably detected) and
(II) automating preprocessing tasks required when analyzing and segmenting large SPT data sets.
The technique introduced is applicable to various scenarios where SPT trajectories are sampled
frequently in time and particles can be accurately tracked over multiple frames, e.g. [15, 16, 23, 26, 28].
Extracting accurate and reliable force estimates from noisy position vs. time data in the aforementioned
setting requires one to account for numerous complications inherent to experimental SPT data in living
cells. For example, nonlinear and/or time changing systematic forces need to be differentiated from
thermal fluctuations (i.e., random forces), both of which contribute to motion at the length and time
scales measurable in living systems [23, 24, 31]. Furthermore, additional measurement noise (consisting of
localization error amongst other factors [18, 23, 33–35]) induced by the optical measurement apparatus
must be systematically accounted for since this noise source varies substantially between and within
single trajectories; inaccurate effective measurement noise estimates can appreciably influence estimates
of kinetic parameters as well as statistical decisions about the underlying physical system [18, 23, 28, 36].
Finally, extracting forces from position vs. time data requires one to explicitly or implicitly make
numerous assumptions about the underlying effective dynamics. We believe these assumptions should
be systematically tested directly against experimental data before one trusts kinetic quantities inferred
from experimental data [23, 37, 38]. However, in live cell SPT studies, reference “ground-truth” is rarely
available. Hence, techniques checking statistical assumptions directly against data are attractive (e.g.,
through goodness-of-fit hypothesis testing [23]).
The HDP-SLDS approach combines Hidden Markov Modeling (HMM), Kalman filtering, [39, 40], and
more recent ideas from Dirichlet Process modeling [41]. We demonstrate how the HDP-SLDS method can
be used to reliably identify the time at which a state change occurs as well as the number of states implied
by a specific time series. In the HDP-SLDS approach [32], the number of underlying states are inferred
from the data via nonparametric Bayesian techniques [41, 42]. Inferring the number of states jointly with
the parameters determining the dynamics (i.e., in a single fully Bayesian computation) is useful because the
number of underlying effective states is rarely known a priori in live cell SPT applications due to inherent
heterogeneity between and within trajectories [21–23]. Since the HDP-SLDS framework directly infers the
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number of states from observed data, the user does not need to provide an accurate upper bound on the
number of states or worry about a posteriori model selection issues [21,43,44]. This allows the HDP-SLDS
approach to readily identify a wide range of distinct dynamical regimes which may occur within a single
trajectory. Once the experimental trajectory is segmented into distinct kinetic states, one can use classic
maximum likelihood estimation (MLE) techniques to infer kinetic parameters [23, 45]. Carrying out
MLE estimation after segmentation mitigates the bias induced by prior assumptions. It is emphasized
throughout that the procedure described in this work can systematically account for measurement noise
and complex spatial-temporal variations in thermal fluctuations & force/velocity fields.
In the particular application that we studied, the heuristics suggested for generating HDP-SLDS prior
parameter input required modification before accurate results could be obtained in SPT applications.
Specifically, prior specification heuristics employed by the original HDP-SLDS Bayesian analysis (using
guidelines outlined in Ref. [32]) required substantial modification (prior specification substantially influences
both state inference and molecular motion parameter estimation). In particular, the “hyperparameters”
governing the measurement and thermal noise (parameterizing components of the “base-measure” [32, 46])
required more careful calibration/tuning before high accuracy results could be obtained. In S1 Text, we
discuss how ideas in Refs. [18, 23] can be used for this type of hyperparameter tuning.
We note that the experimental applications are meant to serve as a proof-of-concept. The results
presented are not intended to provide an exhaustive study of the forces involved in chromosome movement
in metaphase. The problem is amenable to the HDP-SLDS approach since the GFP tagged experiments
experienced abrupt state changes in the experimental trajectories before other downstream SPT computations could be reliably carried out (e.g., force and diffusion coefficient estimation). Not all “abrupt”
were visually obvious to human observers, hence the approach is promising in both data-mining and
preprocessing contexts. It should also be noted that two recent articles have discussed the utility of
nonparametric Bayesian ideas in single-molecule data analysis [47, 48]; Ref. [47] is similar in spirit to this
manuscript except the focus is exclusively on simulation data mimicking features of SPT data and Ref. [48]
focuses on using nonparametric Bayesian modeling to detect the number of states in HMM models. Note
that in Ref. [48] each observation is assumed to be drawn independently from a distribution indexed
by a latent state and the approach employed does not assume explicit temporal dependence between
successive SPT time series observations whereas this work accounts for such temporal dependence (allowing
quantitative approximations of spatially or temporally dependent effective forces and/or velocities).
This article is organized as follows: the mathematical models and experimental methods used are
presented in Sec. 1; Results obtained when analyzing chromatid dynamics in yeast are presented in Sec. 2.
Conclusions are presented in Sec. 3. We have also included Supporting Material (S1 and S5 Texts) where
additional technical details are provided and control simulation studies are analyzed (results shown in
S1-S7 Figs.). A companion publication [47] compares the HDP-SLDS approach to an open-source SPT
Hidden Markov Model assuming a finite number of states through simulation studies.
4
1
1.1
Materials and Methods
Background and Models Considered
We begin by reviewing the main technical ideas underlying the HDP-SLDS introduced in Ref. [32]. Note
that the SLDS models assumed allow researchers to decouple noise induced by thermal fluctuations,
measurement noise, and spatially anisotropic velocity/forces. Forces are computed by leveraging the
overdamped Langevin model [23] (this relationship is made explicit in Sec. 1.1.1 and S3 Text).
In the early SPT works, the spatial and temporal resolution afforded by the measurement device
led researchers to focus mainly on Mean-Square-Displacement (MSD) type analyses to analyze singlemolecule data [49–52]. MSD approaches have many undesirable features, namely they tend to introduce
unnecessary temporal averaging (i.e., they ignore the natural time ordering of the trajectory measurements)
and they have a difficult time accounting for spatially varying forces (a common occurrence in live
cells [23]). Advances in spatial and temporal resolution have inspired many researchers to develop new
techniques aiming at more reliably extracting single-molecule level information out of measurements
[19, 21, 22, 27, 36–38, 44, 53–55]. The previously cited works are most similar in spirit to the work
presented, but all of the works encounter technical difficulties when the number of “states” is not known in
advance. Additional technical complications arise when position estimates are obscured by non-negligible
“measurement noise” [18, 37] or force/velocity fields exhibiting temporal dependence [23].
The method of Fox et al. [32] overcomes the difficulties mentioned above by assuming that a discrete
time series model of the form:
~ri+1 =~µ + F ~ri + ~ηi ; ~ηi ∼ N (0, Σ)
~i+1 =~ri+1 + ~i+1 ; i+1
ψ
~ ∼ N (0, R),
(1)
(2)
can be used to describe the dynamics of each unique state. The position of the molecule or particle at time
ti is denoted by the vector ~ri and the measured value of the position at this same time is denoted by ψ~i
(subscripts are used to index time); the position is not directly measurable due to “localization noise” and
other artifacts induced by the experimental apparatus. The term “effective measurement noise” is meant
to include the net measurement noise induced by finite photon counts, background fluorescence, motion
blur, etc. [18, 23, 35, 37]. Effective measurement noise is modeled as a mean zero normal random variable
with covariance R; the expression ~ ∼ N (0, R) conveys that the random vector, ~, is distributed according
to the normal distribution N (0, R). Techniques for checking the validity of this modeling assumption via
hypothesis testing technique are discussed in Ref. [23].
The term ~µ represents a (constant) “velocity vector” experienced by the particle multiplied by the
observation time ∆t; the matrix F accounts for systematic spatial variations in forces or velocity; random
thermal fluctuations are modeled by ~η . The “drift terms” [56, 57], i.e. ~µ and F ~ri , can be used to quantify
active (linear) force and velocity fields; spatial anisotropy in the force and velocity fields are accounted for
by F ~ri . The physical assumptions we employ for inferring forces from position vs. time data are discussed
in Sec. 1.1.1. The net parameter vector characterizing the (discrete) dynamics of a single state is given
by θ = (~µ, F , R, Σ). Note that the HDP-SLDS presented in Ref. [32] assumed that all observations are
uniformly spaced by ∆t time units.
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Although the molecular position, ~r, is not directly observable, the discrete state-space model above
allows one to readily plug directly into the established Kalman filtering equations [39, 40] to infer the
~ The Kalman filtering framework allows one to systematically treat
dynamics of ~r using the observations ψ.
measurement noise as well as spatially dependent particle forces and/or velocities (this spatial variation
can induce classic “confinement” or “corralling” effects or be indicative of molecular binding in SPT
data [23, 36]). The importance of properly accounting for measurement noise is demonstrated in S2 Text
and is also presented elsewhere [18, 23, 36, 38].
~ and (unobservable) ~r is shown in Fig. 1;
An illustrative trajectory of the (observable) measurements ψ
this figure will also be used to also illustrate the “latent state” modeling discussed in the next paragraph.
Note that the presence of F slightly complicates physical interpretation of the parameters [23], but the
discrete model above can be readily mapped to a continuous time stochastic differential equation (SDE)
studied in Ref. [23] (where diffusion coefficients, effective friction, and instantaneous force terms associated
with an overdamped Langevin equation can be readily extracted). The equations mapping discrete time
parameters used in the HDP-SLDS framework to continuous time SDE parameters are presented in S3
Text for the reader’s convenience. The ability to map between discrete and continuous time models is
important for both specifying physically inspired priors and for interpreting parameter estimates since
classical quantities like diffusion coefficients and forces are typically defined via continuous time SDE
models [17, 18, 23].
In the discrete time model above, θ contains the parameters (vectors and matrices) required to specify
the stochastic dynamics defining the evolution equations of the system in a single “state”. However,
simple linear evolution equations are not expected to be valid for the entire duration of the trajectory in
live cell measurements [23]. In this article, we assume that θ can change abruptly over time or space and
hence the system’s “state” can change over time. We will use the notation zi to denote the state at time
i and will use θz to denote the parameter vector characterizing the dynamics of ~r when it is in state z.
The standard goal of an HMM inference procedure is to infer the state sequence {z1 , z2 , . . . zT } from an
~1 , ψ
~2 , . . . ψ
~T }.
observation sequence {ψ
The typical HMM framework assumes temporal transitions between the states are governed by
Markovian transition probabilities. If there are K fixed states, the transition vector associated with state
(i)
(i)
(i) i in the traditional HMM framework is prescribed by the vector ~π (i) = π1 , π2 , . . . πK . Note that
each component provides the probability of the state transition from state i to state j and the sum of
components of this K dimensional vector is one (hence defines a proper discrete probability distribution);
the collection {~π (1) , ~π (2) , . . . ~π (K) } defines the classical HMM “transition matrix”. In Bayesian inference
of standard HMMs, the Dirichlet distribution is sometimes used as a prior for ~π (i) [21, 55]. Additional
details on the Dirichlet distribution and various infinite dimensional extensions, namely the Dirichlet
Process and the Hierarchical Dirichlet Process, are provided in S4 Text.
The HDP-SLDS framework of Fox et al. comprehensively addresses issues not accounted for in
Refs. [21–23, 27, 58]. The following modified quote from Ref. [32] captures the essence of the HDP-SLDS
method: “The HDP-SLDS is an [infinite discrete state space] extension of hidden Markov models (HMMs)
in which each HMM state, or mode, is associated with a linear dynamical process.” As in Ref. [23],
the statistical influence of thermal fluctuations and effective measurement noise on the dynamics is
accounted for using the Kalman filter framework (this allows linear spatial variations in force and/or
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velocity); however, the HDP-SLDS also provides a mechanism for segmenting trajectories into chunks
where dynamics are distinct. In addition, the HDP-SLDS framework advocated in Ref. [32] introduces a
“sticky parameter” encouraging temporal state persistence; this feature can be advantageous in many SPT
applications [47]. The ability to avoid model selection nuances [21, 43, 44] and infer the number of states
in a data-driven fashion within a single Bayesian inference while also rigorously accounting for spatially or
temporally varying stochastic dynamics are major advantages of the HDP-SLDS method.
Before discussing the major technical weaknesses of the HDP-SLDS method, we discuss some
strengths/weaknesses of published SPT modeling approaches. Wavelet methods have the ability to
identify sharp and abrupt changes in time without making too many assumptions about the underlying
stochastic process [22], but they do not readily account for subtle spatial variations in the dynamics or
noise (e.g., temporally varying velocity fields or diffusion coefficients) and can have difficulty in separating
diffusive noise from measurement noise. The HDP-SLDS overcomes the aforementioned challenges by
assuming a specific parametric model (however, if anomalous diffusive noise is deemed important to
quantify system dynamics [22, 26], this can be a problem for the HDP-SLDS approach). Ref. [58] is one
of the pioneering efforts attempting to account for spatial and temporal variations in SPT signals, but
the approach neglected to explicitly account for the effects of measurement noise (the approach also
focused on comparative hypothesis tests and model selection as opposed to goodness-of-fit tests directly
checking the consistency of a model’s distributional assumptions to experimental data). The method in
Ref. [27] attempts to account for spatial variations (allowing for correlated 2D forces) and the influence of
measurement noise, but appeals to ad hoc statistical approximations which can adversely affect state and
parameter inference [18]; these approximations (which can be avoided using Kalman filtering ideas [23])
can substantially complicate downstream analysis where one would like to check the assumptions against
experimental data [36] (also Ref. [27] did not employ any formal hypothesis testing procedures). The
“Windowed local MLE” method in Refs. [37,45] can account for spatial variations in force and measurement
noise [23, 28] and provides a procedures for goodness-of-fit testing, but does not prescribe a systematic
and automated method for segmenting time series data into distinct dynamical segments.
Open problems facing the HDP-SLDS are associated with prior specification and model validation [46].
The priors used are selected primarily for computational convenience. For example, the so-called matrix
normal inverse-Wishart (MNIW) and other priors proposed allow for exact Markov chain Monte Carlo
(MCMC) sampling [32], but such priors have no real connection to the physical mechanisms governing
molecular motion. The priors and their associated hyperparameters used by the HDP-SLDS method are
discussed further in S5 Text. In the results, we demonstrate how a variant of the method in Ref. [23] can
be used to correct for some artifacts introduced by “bad priors” (i.e., the parametric prior distributions
assumed do not accurately reflect the true underlying process). Quantitatively prescribing a “good prior”
is difficult because single-molecules trajectories contain a high degree of heterogeneity induced by the local
micro-environment, conformational fluctuations of the biomolecules, etc. Hence finding an accurate prior
representative of a single trajectory is non-trivial (finding a prior governing a population of trajectories
in the spirit of Ref. [55] is even harder in SLDS modeling). Fortunately, the nonparametric Bayesian
HDP-SLDS from Ref. [32] combined with frequentist ideas [23] provides researchers in SPT analysis a set
of tools which can be used to more reliably and systematically extract information from complex in vivo
measurements as we demonstrate in the Results and Supporting Text.
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1.1.1
Inferring Instantaneous Forces from Noisy Position Measurements
In what follows, we present a technical discussion explicitly pointing out how we infer forces from a
sequence of position measurements. Particle position is modeled using the overdamped limit of the
Langevin equation (i.e., a first order SDE). The general form of the corresponding first order SDE is:
d~rt = γ −1 f~(~rt )dt +
√
~ t,
2DdB
(3)
where γ is a local effective “friction matrix” associated with the Langevin model and f~(·) represents the
force vector [23, 27]; within the SLDS the force is a function of both the position ~r and the latent state.
The notation above was selected to facilitate comparison to Refs. [27] and [59]. The above model assumes
particle inertia can be ignored (i.e., the dynamics are in the “overdamped” regime or the Smoluchowski
limit of the full Langevin equation [59]) and local diffusivity is constant. In Text S3, the mapping
between the continuous and discrete HDP-SLDS time series models is provided (note that in S3 Text,
γ −1 ≡ Φ). On the time and length scale of SPT measurements, it is generally accepted that particle
inertia is unimportant. However, goodness-of-fit tests can be used to check if unobserved and unmodeled
momentum dynamics affect position measurements [28, 60]. If inertia is believed or empirically determined
to be important, more complex (second order SDE) models for fitting forces from position data can be
entertained [61].
The subtler assumption implicit in our estimate of the forces from the above model is that the Einstein
relationship can be used to relate local effective friction to the local effective diffusion coefficient via the
relationship:
γ −1 =
D
.
kB T
(4)
The above relationship is a classic microscopic relationship relating molecular diffusivity to local friction
[59, 62, 63]. On longer time scales, the microscopic parameters become effective parameters [23, 59] and the
validity of the above relationship can become questionable due to phenomena like molecular crowding [64]
or other factors (e.g., transient binding) inducing more complex dynamics. The temporal segmentation
approach proposed aims to detect when a sudden event alters the dynamics in a statistically significant
fashion, hence the odds of “mixing” different dynamical regimes is reduced (e.g., if the obstacle density
experienced in one part of the cell is vastly different than another, then the effective dynamics in each
regime will be different [64]). The HDP-SLDS’s ability to identify regimes, within a single trajectory,
where locally linear SDE models are consistent with the data makes the validity of Eq. 4 more plausible.
It should be noted that “effective force” data has been backed out from noisy position measurements in
multiple in vitro single-molecule manipulation experiments [37, 38, 57]. In the aforementioned works, force
measurements obtained via an external probe (e.g., AFM) could be compared to internal forces inferred
from noisy position vs. time data (more precisely molecular extension vs. time) obtained by appealing
to relationships similar to those in Eqns. 3 and 4. Despite the system’s being far out of equilibrium
in Refs. [37, 38, 57], appealing to the relationship in Eq. 4 yielded effective “internal forces” consistent
with those measured by external probes. However, testing the validity of the above expression directly
against in vivo SPT data is non-trivial since directly measuring fores in the crowded and heterogeneous
environment of live cells is difficult. The formal statistical hypothesis testing procedure employed can only
8
judge the consistency of the assumed drift and diffusion terms of the SDE in Eq. 3 against observational
data; said differently, only the product γ −1 f~(~rt ) affects the test (the validity of Eq. 4 does not influence
the hypothesis test in any way). As we will discuss later, our force estimates obtained (without assuming
or inputting a local effective cellular viscosity) are consistent with other in vivo force measurements [65].
Even if a researcher does not believe Eq. 4 should hold, our fitted models can still produce local effective
velocity fields [59] which can aid in our quantitative understanding of intracellular dynamics.
Before proceeding, we explicitly mention a few more technical notes about our nominal force estimates.
Our models make no assumptions about the existence of a gradient potential field [27] or the stationarity
(temporal or spatial) of statistics characterizing the random process [59]. Transient structures and
phenomena, which are often resolvable in live cell SPT measurements [23], can invalidate the validity of a
force obtained by taking the gradient of an energy potential field and can also complicate nonparametric
estimators [59] (nonparametric estimators often make implicit stationarity assumptions). Our models
sample data on millisecond timescales, but allow for the local effective forces and diffusion to change over
time and/or space (i.e., statistics are allowed to be non-stationary); we believe this feature will be critical
in accurately characterizing heterogeneous high resolution optical microscopy data.
1.2
Experimental Methods
We examined the in vivo dynamics of chromatin during mitosis to determine the behavior of a region
of the chromosome visualized through the binding of lac repressor fused to GFP (LacI-GFP) to lac
operator (lacO). The lacO is a repeated array of operator DNA sequences (256 repeats, 10 kilobase prs.)
integrated 6.8 kb from the centromere on chromosome XV. The spindle pole bodies (sites of microtubulare
nucleation) are visualized through a fusion protein between a spindle pole component (Spc29) and RFP
(red fluorescent protein). Cells were grown to logarithmic phase at 24 ◦ C in rich media. Images were
acquired on a Nikon Eclipse Ti wide-field inverted microscope with a 100x Apo TIRF 1.49 NA objective
(Nikon, Melville, New York, USA) and Andor Clara CCD camera (Andor, South Windsor, Connecticut,
USA) using continuous laser illumination. Images were streamed at the net effective camera acquisition
rate of 22 frames/sec. The CCD camera’s exposure time was set nominally to be the inverse of the net
frame rate. Images were acquired at room temperature with Nikon NIS Elements imaging software (Nikon,
Melville, New York, USA). The program “Speckle Tracker” and other methods outlined in Ref. [66] were
used to estimate the centroid of GFP and RFP spots for position measurements.
2
Results
In Fig. 2 we display a histogram of the estimated instantaneous force magnitudes as well as the force
magnitude trajectory computed from a time series of measured X/Y chromatid positions (the X/Y data
and state estimates are shown in Fig. 3). The force was estimated by first determining the number of states
implied by the observed trajectory and using the method described in S1 Text to segment the trajectory.
For the experimental trajectories shown in this paper, the initial prior mean of the measurement noise
covariance matrix, R, was assumed to be the identity matrix multiplied by a scalar, σ 2 , where σ = 40nm.
This value was inspired by the fact that the effective measurement noise standard deviation was found
to be in the 10 − 60nm range for these SPT trajectories (with a mode at 40nm). We specified K = 10
9
for the so-called “weak limit approximation” of the HDP-SLDS [67]; this term is discussed further in
S5 Text and the relative insensitivity of the HDP-SLDS method to the K parameter is demonstrated
in Fig S5. Additional parameters required to run the HDP-SLDS segmentation are reported in S5 Text.
After the HDP-SLDS segmentation was obtained, the estimated MLE parameter vector of each unique
state was used to obtain molecular position estimates via the Kalman filter [32, 39]. Finally, the position
estimates along with the MLE parameters were plugged into the equations shown in S3 Text to evaluate
the instantaneous effective force (note: this provides a collection of 2D force vectors).
The force vector magnitudes reported in Fig. 2 observed are representative of other chromatid data sets
studied using this analysis. However, some datasets exhibited more interesting “force regime changes” (as
shown and discussed in the final experimental SPT trajectory studied). Two things should be emphasized
in the analysis of this trajectory: (I) previously published works have reported that chromatids in
metaphase-like conditions experience forces in the (relatively low) 0.1-0.2 pN range [65, 68]. Our findings
are consistent with these previous results, except the model considered here does not require an estimate
of the effective viscosity (a quantity difficult to estimate in live cells) to infer force vectors. Local effective
forces are estimated using time series analysis techniques applied to the so-called overdamped Langevin
equation outlined in Sec. 1.1.1. (II) The HDP-SLDS method aided in accurately determining a transition
between two states (physical interpretation of the identified states discussed below).
The top left panel of Fig. 3 displays the white light image of a trajectory obtained from the yeast
chromatid SPT experiments. Note that we use the phrase “white light image” throughout to indicate a
single diffraction limited image of the yeast cell obtained with white light illumination; the trajectory
obtained using a laser frequency tuned to enhance GFP excitation was then overlaid upon this single
image (all images are recorded at different times, but the yeast cell is not expected to move substantially
during the experiment, so the white light illumination image gives one an idea of the spatial environment
explored by the molecule). The top right panel displays the trajectory in X/Y space with time information
color coded. The red vertical line in the bottom left panel of Fig. 3 shows the time at which a state
change point was detected by the HDP-SLDS method. The bottom right panel of Fig. 3 shows that
state estimates of the vbSPT [21] and HDP-SLDS methods both identified two states, though the latter
captures the state persistence more accurately.
After the temporal segmentation was formally estimated using the HDP-SLDS, we computed the
posterior mode of θ implied by the assumed HDP-SLDS model for each of the states in the two identified
temporal segments. These two parameter vectors (one vector drawn for each of the two states) were used
along with the experimental data to compute the Q test statistic [69] and the corresponding p−values.
The null hypothesis of the one-sided goodness-of-fit test using the Q statistic assumes that the time
series was generated by a Markovian SDE (evaluated at an “optimal” parameter estimate [69]) and
the alternative hypothesis assumes that the time series was produced by any other process [23, 60, 69].
“Optimal” parameter estimates were obtained by two means: (i) drawing the parameter vector obtained
after 104 iterations of the MCMC sampler in the HDP-SLDS inference; note that this approach included
prior bias and (ii) via MLE estimation applied to HDP-SLDS segmented trajectories; with this approach,
bias is mitigated since priors do not influence the parameter estimates.
In the first segment, using the observed data and the HDP-SLDS parameter vector estimate mentioned
above, resulted in a p−value of 0.32 (little evidence for rejection), but in the second segment a p−value <
10
0.01 was computed (large evidence for rejection). Visual inspection of Data S1 (see rightmost spot) shows
the effective measurement noise has abruptly changed, however the prior assumed by the HDP-SLDS
segmentation only allowed the components of R to change modestly. The default settings of the MATLAB
code associated with Ref. [32] assumes little dispersion about the nominally known mean R (all segments
yielded an effective measurement noise having ≈ 40.0nm on all diagonal components via the HDP-SLDS
method). Using a so-called “non-informative” or “uniform” prior can potentially remedy the situation,
but such a prior can introduce both technical and computational problems in nonparametric Bayesian
methods [42]; see S5 Text for further discussions on this issue.
Alternatively, if one uses the segmentation afforded by the HDP-SLDS, but instead computes the
MLE using a variant of the technique of Ref. [23] (discussed in S1 Text) as opposed to using the posterior
sample provided by the HDP-SLDS (which contains biases induced by priors), the p−values were found to
be 0.31 and 0.45 for the left and right segments, respectively. The primary difference in the estimation
results was in the noise components estimated, R and D, the effective measurement noise and diffusion
coefficient matrices, respectively (see S3 Text). For example, the MLEs R̂ = diag([19.52 , 39.92 ]) [nm2 ] &
D̂ = diag([1.2 × 10−1 , 1.4 × 10−2 ]) [µm2 /s] were obtained in the left segment and R̂ = diag([140.12 , 93.92 ])
[nm2 ] & D̂ = diag([4.8 × 10−5 , 4.5 × 10−5 ]) [µm2 /s] were obtained in the right segment (i.e., the segment
occurring later in time) using the model in Ref. [23]. Hats are used to denote MLE estimates and diag(·)
denotes the square diagonal matrix formed by the arguments. Note that the diffusion coefficient estimate,
D̂, is near machine single-precision zero in the right segment.
The GFP image stack associated with this video readily shows photobleaching occurs (see rightmost
point spread function spot in S1 Data). The MLE predicted effectively zero diffusion for the right
segment (i.e., the state appearing temporally after the first segment), quantitatively indicating that
significant photobleaching had occurred and that the second state was mainly background photon noise
not corresponding to an individual molecule. The quantitative evidence for this statement is the MLE
parameter estimates (effective measurement noise near diffraction limit in the “right segment”) and the
corresponding p−values, reported in two paragraphs above, resulting from testing the fitted model against
the data. In this experiment, the HDP-SLDS approach was able to automatically detect this measurement
noise transition despite a poorly tuned prior; the state change was identified without requiring subject
matter expertise or manual image inspection.
Fig. 4 displays another experimental trajectory with a different type of state switching which was
readily identified by the HDP-SLDS procedure. The posterior mode computed using Gibbs sampling of
the HDP-SLDS model contained assignments with two and three states; in total 104 posterior MCMC
samples were sampled. A majority of the MCMC draws implied two-state switching. However, one does
not necessarily need to be concerned about the number of states if one is only interested in determine
change points. The change points predicted by the HDP-SLDS are denoted by vertical lines (change
points were determined to occur if a state change occurred at time i in over 25% of the posterior samples).
Accurately identifying change points can help in quantifying the transition times between distinct physical
states [21]. However the state switching in this particular system was likely caused by vibrations in the
microscope’s piezo stage. The HDP-SLDS can be used to automatically identify physically relevant state
changes or state changes induced by experimental artifacts. As we show in Fig. S5, isolated outliers (e.g.,
caused by intense fluorescent background “flashes” in some SPT experiments) are also readily identified
11
by the HDP-SLDS framework.
Next we turn to an example exhibiting subtle state changes (identified using the procedure outlined
in S1 Text) in a more biologically relevant study. Fig. 5 shows two GFP tagged chromatid trajectories
during the metaphase stage of mitosis. The white light image of these trajectories is shown in Fig. 6
where the spatial location of the RFP tagged mitotic spindle pole bodies (SPB) as well as the GFP
tagged chromatid trajectories are shown. HDP-SLDS state segmentation was carried out for each of the
trajectories; the state segmentation is denoted by vertical lines in the bottom panel of plots in Fig. 5.
The procedure used to infer forces in Fig. 2 was carried out again for the pair of chromatids undergoing
mitosis. The histogram of the force magnitudes is not atypical for trajectories observed, however the
evolution of the force vectors was suggestive of a dynamic regime shift.
In order to probe this regime shift further, we computed the eigen-values and eigen-vectors of the
MLE of B (a matrix associated with the continuous-time overdamped Langevin model; details provided
in S3 Text). The eigen-analysis shown in Fig. 6 provides quantitative information about variations in
the effective force vector as a function of position (i.e., the eigen-analysis provides both magnitude and
directional information about a linear vector field). The upper-right panel shows that the largest magnitude
eigen-value consistently decreased over time for each state identified by the HDP-SLDS segmentation. The
rate of decrease was similar for the pair of sister chromatids. This phenomenon was originally identified
using the crude time window segmentation of Ref. [23], but the HDP-SLDS provides a framework for
more systematically dividing the trajectory into different dynamical segments.
In terms of physical interpretations, several molecular components can induce “force cross-talk”
between sister chromatids. A potential explanation of the eigen-value phenomenon described above is
that microtubules of the mitotic spindle connected to the common centromere of the sister chromatids are
simultaneously inducing a common force change to both chromatids. Alternatively (or perhaps in addition),
a common tension signal induced by chromatin loops and a network of pericentric proteins (cohesin and
condensin [24, 70]) might be influencing the forces experienced by the pair of sister chromatids; recall that
the GFP dyes used are in close proximity to the centromere so molecular changes in the pericentric region
can potentially influence the fluctuations observed in the tagged sister chromatids [24, 70].
Regardless of the underlying biological cause of the phenomenon observed, the downstream eigenanalysis substantially benefited from the temporal segmentation provided by the HDP-SLDS framework.
Using the segmentation afforded by the HDP-SLDS method resulted in the eigen-vectors shown in the
bottom panel of Fig. 6; in this figure, the eigen-vectors associated with the largest magnitude eigen-value
is displayed as a solid line and the eigen-vector associated with the smaller eigen-value of the estimated 2D
B matrix is shown by a dashed line (the origin for the eigen-vectors correspond to the empirical average of
the identified states). The (real-valued) eigen-values and eigen-vectors provide spatial information about
restoring force directions experienced by the tagged chromatid. Observe how the pair of chromatids both
exhibit a state change occurring around 16s. Note that the time of the change point occurring around
16s coincides for the sister chromatids, but each trajectory was processed independently by separate
HDP-SLDS analyses. At 16s, the dominant eigen-vectors of the chromatids change from pointing towards
the spindles to a direction nearly orthogonal to the plane connecting the two mitotic spindle poles. The
eigen-values for both chromatids are decreasing at roughly the same rate (i.e., their effective “stiffness”
is increasing). Also, recall that the pair of chromatids are subject to forces from multiple sources that
12
can change in magnitude and direction; for example, tugging induced by the microtubules connected
to the centromere shared by the chromatids, forces associated with nucleosome remodeling (e.g., RNA
polymerase “tugging” on the strand as it makes RNA), as well as other spring like forces induced by the
shared centromere and percentric components [24, 70].
The various explanations posited above are admittedly speculative. What these results unambiguously
show is that sister chromatids experience a force regime shift whereby the direction and magnitude of the
forces (inferred by position vs. time data) co-occur in both direction and magnitude. This information
gives new insight into the effective forces experienced by particles in time changing, complex live cellular
environment. Researchers can get a more reliable “force map” that contains rich dynamical information
beyond that contained in a mean square displacement or correlation-based analysis. Note that this “force
map” allows a spatially dependent vector field to change over time (an important feature for detecting
dynamical changes in single-molecule measurements). A dominant force direction pointing orthogonal to
the spindle axis is highly suggestive of a new dominant force (not induced by the microtubule pulling) being
felt by both sister chromatids. Deciding the correct physical explanation of the phenomenon observed
requires additional experiments, but the results obtained on the metaphase trajectories are presented
in order to show the power of SPT analysis coupled with HDP-SLDS segmentation and downstream
likelihood-based time series analysis [23]. Currently we are attempting to use this type of analysis to more
systematically and quantitatively probe forces associated with the various physical phenomena described
here using experiments which more carefully control the types of forces that can be experienced by the
GFP tagged chromatids.
Before proceeding, we should note that the GFP array used to tag the chromatids induces a gradual
continuous trend in the effective measurement noise (the finding can readily be observed in S1-S3 Data).
A gradual time trend induced in large part by different fluorophores bleaching out has been observed
and quantitatively studied in previous works (see Supp. Mat. of Ref. [23]). However, the HDP-SLDS
formulation considered assumes a constant measurement noise for the entire duration of the trajectory.
S6 - S7 Figs. explore various simulations mimicking features in the experimental data; specifically we
analyze data where the effective measurement changes linearly from 15nm to 35nm over the course of 400
measurements (this induces model misspecification since the HDP-SLDS model assumes fixed covariance).
The linear change in measurement noise is consistent with the MLE range of values observed in S3 Data.
Figs. S6 - S7 indicate that if one uses a prior near the temporal midpoint of the time changing measurement
noise that the HDP-SLDS segmentation can achieve accuracy similar to the ideal situation where the
HDP priors match the DGP exactly. For whatever system is being studied, it is highly recommended
to employ simulations mimicking features implied by the data in order to be more confident that the
segmentation output does not contain artifacts induced by features not explicitly accounted for in the
underlying model. Techniques accounting for gradual time trends in certain parameters extending the
HDP-SLDS model would be an interesting future research topic.
In terms of more general applicability of this type of analysis, systematically segmenting SPT trajectories
into distinct dynamical regimes and then using classical statistical physics models is one alternative to using
so-called anomalous diffusion models commonly used in SPT analysis [20, 26, 71]. This is advantageous
since fitted overdamped Langevin model output can readily be interpreted in terms of force, velocity,
and molecular friction. Furthermore, an overdamped Langevin SDE model can be tested directly against
13
experimental data and the effects of measurement noise can be systematically accounted for by likelihood
methods [23, 28, 37, 38]. Anomalous diffusion models are more difficult to physically interpret and rigorous
time series analysis (accounting for measurement noise, sampling noise, spatially dependent forces, etc.) is
challenging for various technical reasons [23].
3
Discussion
We have demonstrated the utility of the HDP-SLDS method of Fox et al. [32] in automatically segmenting
SPT data into different dynamical regimes. The approach shows great promise in systematically processing
a variety of SPT data sets. When applied to experimental data, we demonstrated how new quantitative
information can be extracted about transient forces experienced during mitosis in live yeast cells; the
method explicitly accounts for the statistical effects of measurement noise on top of “thermal” or “process”
noise (both sources are non-negligible in many SPT applications). Simulations results reported in S2 Text
were used to exhibit strengths and weaknesses of the HDP-SLDS approach within an SPT context. Work
presented in Ref. [47] compared the HDP-SLDS approach to classic (finite state space) HMM models.
In terms of strengths, the HDP-SLDS approach can account for an unknown number of states, position
dependent anisotropic active forces, and can account for the statistical effects of measurement noise
by exploiting Kalman filtering modeling in conjunction with HDP ideas [41]; these features are not
accounted for in any HMM approach currently used in SPT analysis and hence the HDP-SLDS represents
a “state-of-the-art” technology within this domain. The approach provides a more systematic means
for the “time window size” selection problem discussed in Ref. [23]. The HDP-SLDS approach was
also confirmed to be fairly robust to many modeling assumptions and hyperparameters associated with
HDP modeling [32]. In situations where a collection of transient dynamical responses are experienced
within a single trajectory and the “states” causing the different kinetics are experimentally resolvable,
the data processing procedure discussed here provides an attractive alternative to anomalous diffusion
modeling [20, 23, 26, 36, 71] since the procedure can mitigate artifacts induced by aggregating distinct
kinetic states [20, 23]. Furthermore, the HDP-SLDS approach can be modified to produce output that can
be readily physically interpreted in terms of classic SPT models (see Sec.1.1.1).
A substantial weakness of the HDP-SLDS approach (also discussed in Ref. [46]) is associated with its
dependence on reliable priors for the parameters determining governing the Kalman filter parameters (i.e.,
hyperparameters specifying the “base measure” [46]). The data-driven scheme advocated in Ref. [32] for
selecting the prior mean of R exhibits undesirable properties in many SPT applications (results presented
in S2 Text). A variant of the MLE-based approach of Ref. [23, 37] was discussed (S1 Text) and was
demonstrated (S2 Text) to be capable of providing pilot estimates needed to tune the base measure
hyperparameters priors in SPT applications. The formal hypothesis testing methods utilized in Ref. [23,37]
also allow one to test the validity of a candidate segmentation and assumed dynamical model against
experimental data in situations where reference (or “ground-truth”) data is unavailable.
In summary, combining recent nonparametric Bayesian modeling ideas, like the HDP-SLDS, with
frequentist ideas show great potential in quantitatively analyzing complex SPT data sets. In the future,
we aim to provide user-friendly software making the tools discussed herein readily accessible to the
diverse set of researchers involved in SPT. The development of new, statistically rigorous data processing
14
algorithms capable of reliably extracting information about molecular motion from live cell data is
becoming ever more important as in vivo microscopy techniques continue to improve in spatial and
temporal resolution [10, 13–16].
Acknowledgments
CPC was supported by internal R&D funds from Ursa Analytics and KB was funded by the National
Institutes of Health R37 grant GM32238.
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26
Position / State
Figures
✓1
✓2
✓3
✓1
Observation #
Fig 1. Illustrative example of a Switching Linear Dynamical System (SLDS). The stochastic dynamics of the (simulated)
~ are represented by
particle is denoted by thin red line (i.e., the trajectory of ~
r) and the simulated discrete measurements, ψ,
the grey lines with symbols. The parameter determining the linear dynamical system is denoted by θz . At random times,
the latent state z changes and hence the underlying parameter specifying the dynamics, θz , also changes. Note that θz
models both thermal and measurement noise (parameters defined in Eqn. 1). The latter is crucial for identifying subtle
changes in the underlying stochastic process. For example, the dramatic change in thermal fluctuations shown around
observations 300-400 would be difficult to detect without jointly modeling thermal and measurement noise. To achieve this
joint modeling, one must also account for temporal correlations induced by measurement noise [18, 36, 37].
28
kF k [pN ]
freq
21
Time [s]
kF k [pN ]
Fig 2. Histogram of inferred force magnitudes computed from experimental chromatid position vs. time data. The inset
displays the force trajectory. Forces were computed using HDP-SLDS state segmentation and Kalman filtering (see text for
details). The white light image of cell with the position vs. time data measured from the GFP channel is shown in Fig. 3.
27 22
Y [µm]
Time [s]
1µm
State
X/Y [µm]
X [µm]
Time [s]
Time [s]
Fig 3. Experimental trajectory of GFP tagged chromatid in yeast. Top left panel: White light image where green lines
denote the SPT trajectory measured via the GFP channel. Top right panel: Zoomed snapshot of trajectory shown in white
light image; time is color coded. Bottom left panel: X/Y vs. time for trajectory. Vertical red line denotes the time point at
which the HDP-SLDS inferred a state change. Bottom right panel: HDP-SLDS [32] and vbSPT [21] state estimates.
X/Y [µm]
23
Time [s]
Fig 4. A GFP tagged 2D chromatid trajectory exhibiting rapid two state regime switching. The vertical red lines denote
the change points identified by the HDP-SLDS analysis.
29
24
Fig 5. Analysis of two sister chromatid trajectories in the metaphase of mitosis (each column shows results of one
chromatid). Spatial location of the spindle pole bodies (SPB) and the two sister chromatid trajectories are displayed in Fig.
6. The top row displays the forces inferred using the Kalman filter analysis discussed in the main text. The bottom row
displays the measured trajectories vs. time as well as the time point of the state changes predicted by the HDP-SLDS
approach.
25
34
SPB 2
1
SPB 1
SPB 2
SPB 1
1µm
Time [s]
SPB 1
Y [µm]
Time [s]
SPB 2
X [µm]
Figure 6. Top left panel: white light image of two chromatid trajectories in yeast; spindle pole body (SPB) locations
Fig 6. Top left panel: white light image of two chromatid trajectories in yeast; spindle pole body (SPB) locations denoted
by red. the
Top largest
right panel:
the largest(quantifies
eigen-value (quantifies
effective restoring
force experienced
chromatid) vs.
vs. time; a
by red. Top denoted
right panel:
eigen-value
effective restoring
force experienced
by by
chromatid)
new restoring
force
wasrestoring
inferred
forwas
each
of the
identified
by the by
HDP-SLDS
methods
(three
time;
a new
force
inferred
forunique
each of states
the unique
states identified
the HDP-SLDS
methods
(threeunique
unique states for
SPB 1 and two states for SPB 2). Bottom panel: Eigen-vectors implied for the various states; solid line denotes the
states for spindle 1 and two states for spindle 2). Bottom panel: Eigen-vectors implied for the various states; solid line
eigen-vector associated with the largest eigen-value and the dashed line denotes the weaker eigen-vector. For spindle 1, we
denotes the eigen-vector
associated
with the
largest
eigen-value
and the dashed were
line denotes
theto
weaker
display the eigen-vectors
associated
with state
2 and
3 (state
1’s eigen-vectors
similar
stateeigen-vector.
2, so theseFor
vectors were
omitted for clarity).
spindle 1, we display the eigen-vectors associated with state 2 and 3 (state 1’s eigen-vectors were similar to state 2, so these
vectors were omitted for clarity).